Integration over a sphere of unit radius (the unit sphere) is, by convention, carried out by the integral .intg..intg. F(.alpha., .theta., .psi.)sin.theta. d.theta.d.psi. where .theta. and .psi. are the polar and azimuthal angles, respectively and .alpha. may represent a set of other variables. For many physical systems, the function F(.alpha., .theta., .psi.) cannot be given analytically. Therefore, the integration has to be performed numerically. Numerical integration over the unit sphere involves a certain partition scheme for the unit sphere. A widely used partition scheme is the rectangular spherical grid with n equally spaced values for .theta., and m equally spaced values for .psi.. Although it is simple and popular, the rectangular spherical grid partition scheme is not economical from the viewpoint of computational time because the unit sphere is partitioned unevenly. When the evaluation of F(.alpha., .theta., .psi.) at each orientation takes a considerable amount of computational time such as in computer reconstruction of a powder or a frozen solution magnetic resonance spectrum (EPR, solid sate NMR, NQR etc.), more evenly partitioned schemes are desired and some of these have been considered in computer simulation of magnetic resonance spectra.
Computer simulation has become a common method in magnetic resonance studies, for extracting the spin Hamiltonian parameters from randomly oriented powder spectra or spectra of disordered spin systems, as taught in references (1,2). However, this practice is still largely limited to relatively simple systems of low dimensions in spin space where analytical solutions exist or perturbation theory can be employed (2-5). A typical example is a spin S=1/2 interacting with a number of nuclear spins where the electronic Zeeman interaction dominates other interactions involved and the latter are treated as a second-order perturbation. For spin systems with S.gtoreq.1/2, perturbation theory is generally not applicable because of the presence of other equally important interactions such as the fine structure.
For a system in which a number of interactions with comparable magnitudes are present, the only solution to this problem is matrix diagonalization. That is, all of the interactions are treated with equal importance and the energy matrix is diagonalized in the entire spin space (6-11). This approach can easily become a formidable task even in the modern era of computers as diagonalization of Hermitian matrices is a O(N.sup.3) process where N is the order of the matrix. For example, with a SUN Microsystems Sparcstation 10/30 workstation, diagonalization of a 36 by 36 Hermitian matrix for both eigenvalues and eigenvectors with one of the fastest routines available, RSH (12) or LAPACK (13) routines, takes .about.0.1 seconds.
A major complicating factor in computer reconstruction of randomly oriented powder spectra is anisotropy. If a system has a high degree of anisotropy, as often found in orthorhombic or lower symmetries, the randomly oriented powder spectrum has to be reconstructed from a very large number of single crystal spectra. Otherwise the simulated spectrum shows considerable ripples, computer noise. For example, a rhombic Mn(II) complex with zero-field splittings comparable with the microwave frequency will yield a spectrum spanning a field range of .about.400 mT. Assuming the full widths at half maximum (FWHM) are of the order of 2 mT and that the tolerance of the resonant field position is not to exceed FWHM, then a minimum of 200 steps in .theta. will be required to cover 90.degree. on an average base. By using the "Igloo" partition method (14), discussed later, some 20,000 orientations in the first octant have to be sampled. Normally, a few diagonalizations, say an average of five, are required to obtain the resonant field position for a particular transition for each orientation. For 30 allowed transitions, six from each hyperfine set, there will be a total of three million matrix diagonalizations required for the generation of the randomly oriented spectrum. Given that a single matrix diagonalization takes .about.0.1 seconds on a modern computer like a SUN Microsystems Sparcstation 10/30 workstation, it would thus take 3.7 days to produce a single simulated spectrum. This is obviously impractical.
One known alternative to avoid a large number of single crystal spectrum calculations is through interpolation (6,7,10,15). In these known interpolation methods the eigenvalues and eigenvectors are calculated exactly for only a small number of orientations while for the majority of orientations, an interpolation method is used. This automatically reduces the computing time. A number of interpolation methods have been used in conjunction with different partition schemes (6,7,10,15). Although these methods have been successfully applied to certain specific problems, there is limited generality to these methods.
Some of the known partition methods used in computer simulations are discussed below:
(1) "Apple-peel" method.
This is the simplest method to partition a sphere which involves the division of the polar angle .theta. from 0.degree. to 90.degree. in N equal steps and the azimuthal angle .phi. in M equal steps. Although this method is simple from the viewpoint of implementing an interpolation scheme (6,7) it is wasteful as half of the orientations (lying between .theta.=0.degree. and 45.degree.) cover less than 30% of the total solid angle of the first octant. A modification of the Apple-peel method has also been used by some (2).
(2) "Igloo" method.
The Igloo method, developed by Belford et al. (14), partitions the unit sphere more evenly. An example of this method is shown in FIG. 1(a). There are (N+1) values of .theta. but the number of values for .phi. increases from 1 at .theta.=0.degree. to (N+1) at .theta.=90.degree. in steps of 1. Thus the total number of orientations required in the Igloo method is (N+1)(N+2)/2 and is consequently computationally more economical than the Apple-peel method ((N+1)*M). In FIG. 1(a), N=9, (55 orientations or vertex points). However, this method is unsuitable for the incorporation of interpolation algorithms.
(3) Spiral method.
The spiral method (10) involves setting up a spiral circling about the polar axis (the z axis) of the unit sphere starting at the x axis and spiralling up to the pole. The objective of the spiral method is to keep the arc length between any consecutive points along the spiral constant, and to also keep the separation between the loops along any .phi. constant. The disadvantage of the spiral method is the lack of an analytical solution for determining the next point from a given point, which must be determined from a minimisation algorithm.
(4) Triangular method
Another method used in computer simulation of randomly oriented powder spectra (15) involves an equilateral triangular grid on the faces of an octahedron as shown in FIG. 1(b). (Only one octant is shown). The plane triangles on the faces of the octahedron have the same area, but they do not all subtend the same solid angle. An exact calculation of the solid angles by individual triangles is complicated and thus only an approximation was given.